> but it's a criterion guaranteed to give a unique answer, right?
It's unique.
Message: 6081 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 08:50:23
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> it shows that the 'expressibility' metric is not quite the same
>> as the taxicab metric on the isosceles-triangular lattice. you
>> need to use scalene triangles of a certain type, which i don't
>> think is what you were thinking when you wrote "taxicab
>> complexity" above.
>
> That's true; I just count the rungs. Any lattice that's
> 'topologically' (?) equivalent to the triangular lattice
> will do.
you mean approximately?
>> straightness applies to a set of unison vectors. different sets
>> of unison vectors can define the same temperament. a temperament
>> may look good on the basis of being defined by good unison
>> vectors. but in fact you may end up with a terrible temperament
>> if the unison vectors point in approximately the same direction.
>
> Why would it be terrible?
heuristically speaking,
in most cases the *difference* between the unison vectors will be of
similar or greater magnitude in terms of JI comma interval size as
the unison vectors themselves, but since the angle is very small,
this *difference* vector will be very short (i.e., low complexity). a
comma of a given JI interval size will lead to much higher error if
tempered out over much fewer consonant rungs (i.e., if it's very
short) than if it's tempered out over more consonant rungs (i.e., if
it's long). therefore, you may end up with a temperament with much
larger error than you would have expected given your original pair of
unison vectors.
> Thanks again! Now everything is clear. Except how you
> derrived the heuristics!
i did post the derivation a while back, probably before i even used
the word "heuristic", but if you search for "heuristic", i think
you'll find a post that links to the derivation post.
> Seriously man, one expository blurb would save you from having
> to do this for each person who's interested, and for me again
> when I've forgotten it in 6 months. :~)
i welcome suggestions or just make your own blurb, and let's put this
on a webpage somewhere.
Message: 6082 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:13:49
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>> >an you find them at the end of a map-space search, and take
>> them out?
>
>what do you want to take out?
Can I identify the duplicate temperaments?
>from what?
A search of all possible maps.
> what does "//" mean?
Cut. I've been using it since I've been on these lists.
>correct. the untweakable generator has been tweaked. is that all?
Assuming 2:1 reduction makes me squirm in my chair, is all.
Plentiful near-2:1s should emerge from the search if the criteria
are right.
>> Thanks again. So if reduction is necc., it means that a
>> temperament can be described by two different lists of commas,
>> right?
>
>right, although for single-comma temperaments, only one choice
>leaves you without torsion.
Thanks yet again.
>> This means we'll have the same problem searching comma
>> space as we did map space. So wedgies are our last hope.
>
>the problem i was pointing out with map space, i think, was
>that the arbitrariness of the set of generators means your
>complexity ranking (if it's just based on the numbers of the
>map) will be meaningless.
Oh. Now I get it! You're right. But doesn't the same
problem occur with different commatic representations, when
defining complexity off the commas?
>> Are you saying a badness cutoff is not sufficient to give a
>> finite list of temperaments?
>
>exactly. in *every* complexity range you have about the same
>number of temperaments with log-flat badness lower than some
>cutoff -- and there are an infinite number of non-overlapping
>complexity ranges.
Oh. I guess I need some examples, then, of most of the simple
temperaments that are garbage...
-Carl
Message: 6084 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:18:38
Subject: Re: Mega 11 and 13 limit ets
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>>
>>> what happened to 504504?
>>
>> I checked from 3 to 23, and it's not nearly good enough to make the
>> cut in any of them. Are you sure this is the right number?
>
I'd guess he chose it on the basis of its prime factorization:
504504 = 2^3 3^2 5^2 7 11 13
I'm not sure why he didn't choose the lcm of all numbers up to 15
(360360) or 16 (720720) instead.
Message: 6085 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 09:39:16
Subject: Re: heuristic and straightness
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> A linear regression
>> of log(complexity) vs log(d) gives c ~ .991685*log(d)^.986763,
>
> oh, so you mean to use *your* complexity/badness/whatever as the
> dependent variable in the regression!
>
> clearly, though, you understand the heuristic.
The
heuristic for complexity works particularly well as an estimate for
geometric complexity. I did a regression analysis (which you could no
doubt do better, if you have stat software) which also confirmed that
your heuristic for error seems to have the right coefficients. The
result is that your estimates look like an excellent way of quickly
filtering a list of commas for geometric complexity, rms error, and
geometic badness, which could then be used to compute wedgies. If we
had estimates for the wedgies that would be even better.
> don't forget to report standard error ranges for your coefficient
> estimates!
I don't have a program which gives me residuals or F-ratios or
anything which one really wants to do this right. You can probably do
it better; just remember to take the log of both sides.
Message: 6086 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:22:12
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>> Can you find them at the end of a map-space search, and take
>>> them out?
>>
>> what do you want to take out?
>
> Can I identify the duplicate temperaments?
duplicate? any temperament can be generated from an infinite number
of possible basis vectors.
>> correct. the untweakable generator has been tweaked. is that all?
>
> Assuming 2:1 reduction makes me squirm in my chair, is all.
> Plentiful near-2:1s should emerge from the search if the criteria
> are right.
if the criteria include mapping to, and minimizing error from, 2:1,
then of course a near 2:1 will emerge in each temperament.
>>> This means we'll have the same problem searching comma
>>> space as we did map space. So wedgies are our last hope.
>>
>> the problem i was pointing out with map space, i think, was
>> that the arbitrariness of the set of generators means your
>> complexity ranking (if it's just based on the numbers of the
>> map) will be meaningless.
>
> Oh. Now I get it! You're right. But doesn't the same
> problem occur with different commatic representations, when
> defining complexity off the commas?
not if you define complexity right!
>>> Are you saying a badness cutoff is not sufficient to give a
>>> finite list of temperaments?
>>
>> exactly. in *every* complexity range you have about the same
>> number of temperaments with log-flat badness lower than some
>> cutoff -- and there are an infinite number of non-overlapping
>> complexity ranges.
>
> Oh. I guess I need some examples, then, of most of the simple
> temperaments that are garbage...
what are the 20 simplest 5-limit intervals? now set each of these to
be the commatic unison vector, and what temperaments do you get? gene
has studied a few of them because they actually had low badness --
but the error is so high that the generators themselves sometimes
provide better approximations to a consonance than the interval the
consonance is mapped to!
Message: 6087 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 10:16:06
Subject: Re: heuristic and straightness
From: Graham Breed
--- wallyesterpaulrus wrote:
> straightness applies to a set of unison vectors. different sets of
> unison vectors can define the same temperament. a temperament may
> look good on the basis of being defined by good unison vectors. but
> in fact you may end up with a terrible temperament if the unison
> vectors point in approximately the same direction.
Then that's what you need to reduce the complexity of the search. I
can't find a quantitative definition of "straightness" in the
archives. What is it? I presume it works for insufficient sets of
unison vectors.
Graham
Message: 6088 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:22:22
Subject: Re: Mega 11 and 13 limit ets
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > > Definitions of tuning terms: list of EDOs used... * [with cont.] (Wayb.)
>
> I'd guess he chose it on the basis of its prime factorization:
Sorry, that should be
504504 = 2^3 3^2 7^2 11 13
No 5s, obviously!
Message: 6089 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 10:18:52
Subject: Re: heuristic and straightness
From: Carl Lumma
>> >hat's true; I just count the rungs. Any lattice that's
>> 'topologically' (?) equivalent to the triangular lattice
>> will do.
>
>you mean approximately?
? I originally meant that counting the rungs between a dyad
and the origin would be approximately the same as the log of
the odd limit of the dyad...
lnoddlimit 7limittaxicab ratio
15:8 2.7 2 1.35
5:3 1.6 1 1.6
105:64 4.7 3 1.6
225:224 5.4 4 1.35
>>> but in fact you may end up with a terrible temperament
>>> if the unison vectors point in approximately the same
>>> direction.
>>
>> Why would it be terrible?
>
>heuristically speaking,
>
>in most cases the *difference* between the unison vectors
>will be of similar or greater magnitude in terms of JI comma
>interval size as the unison vectors themselves, but since
>the angle is very small, this *difference* vector will be
>very short (i.e., low complexity). a comma of a given JI
>interval size will lead to much higher error if tempered out
>over much fewer consonant rungs (i.e., if it's very short)
>than if it's tempered out over more consonant rungs (i.e.,
>if it's long). therefore, you may end up with a temperament
>with much larger error than you would have expected given
>your original pair of unison vectors.
Right, the difference vector has to vanish, too. Ok. What
I don't get is, for a given temperament, can I change the
straightness by changing the unison vector representation?
If so, this means that badness is not fixed for a given
temperament...
Also, can I change the straightness by transposing pitches
by uvs?
Finally, is "commatic basis" an acceptable synonym for
"kernel"?
>i did post the derivation a while back, probably before i
>even used the word "heuristic", but if you search for
>"heuristic", i think you'll find a post that links to the
>derivation post.
I think I remember it coming out, but I couldn't find it
today in my searches. I did try.
>i welcome suggestions or just make your own blurb, and
>let's put this on a webpage somewhere.
Maybe the original exposition can just be updated a bit, and
then monz or I could host it, certainly.
-C.
Message: 6090 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:24:05
Subject: Re: Mega 11 and 13 limit ets
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
>>
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
>> >>
>>>> what happened to 504504?
>>>
>>> I checked from 3 to 23, and it's not nearly good enough to make
the
>>> cut in any of them. Are you sure this is the right number?
>>
> I'd guess he chose it on the basis of its prime factorization:
>
> 504504 = 2^3 3^2 5^2 7 11 13
that wouldn't make sense, since he also chose 15601 as "most
convenient" (this time for 3-limit), and it's prime!
Message: 6091 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 10:22:37
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Graham Breed <graham@m...>"
<graham@m...> wrote:
> --- wallyesterpaulrus wrote:
> Then that's what you need to reduce the complexity of the search. I
> can't find a quantitative definition of "straightness" in the
> archives.
i haven't come up with one yet.
> I presume it works for insufficient sets of
> unison vectors.
insufficient? how can a set of unison vectors be insufficient?
Message: 6092 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:25:26
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
> <clumma@y...> wrote:
>>>> Can you find them at the end of a map-space search, and take
>>>> them out?
>>>
>>> what do you want to take out?
>>
>> Can I identify the duplicate temperaments?
>
> duplicate? any temperament can be generated from an infinite number
> of possible basis vectors.
oops -- i meant sets of possible basis vectors, assuming the
temperament is 2-d or more.
Message: 6093 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 10:26:40
Subject: Re: heuristic and straightness
From: Graham Breed
wallyesterpaulrus wrote:
> insufficient? how can a set of unison vectors be insufficient?
One unison vector is insufficient to define a 7-limit linear
temperament, two or three unison vectors are insufficient to define a
21-limit linear temperament. For an arbitrary search to be practicable,
there has to be a way of rejecting sets of three unison vectors because
you know they can't give a good linear temperament. That would reduce
the search to more like O(n**3) in the number of unison vectors instead
of O(n**6).
Graham
Message: 6094 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:45:24
Subject: Re: Nonoctave scales and linear temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
>> Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42 only
> gives
>> 5 million or so combinations.
>
> still vastly redundant.
One
possibility would be to choose a comma list with a badness cutoff,
where the badness was the badness of the corresponding codimension
one, or one-comma, temperament. I think I'll try that unless someone
has a better suggestion.
Message: 6095 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 10:28:00
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
> Right, the difference vector has to vanish, too. Ok. What
> I don't get is, for a given temperament, can I change the
> straightness by changing the unison vector representation?
yes.
> If so, this means that badness is not fixed for a given
> temperament...
that's not true. since both the defining unison vectors *and* the
straightness change, the badness can (and will) remain constant.
> Also, can I change the straightness by transposing pitches
> by uvs?
this is meaningless, as we're talking about temperaments, not
irregular finite periodicity blocks. we're talking either equal
temperaments or infinite regular tunings.
> Finally, is "commatic basis" an acceptable synonym for
> "kernel"?
no. every vector that can be generated from the basis belongs to the
kernel, but not every set of n members of the kernel (even if they're
linearly independent) is a basis (since you may get torsion).
Message: 6096 - Contents - Hide Contents
Date: Sat, 18 Jan 2003 22:51:43
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>> in most cases the *difference* between the unison vectors
>>> will be of similar or greater magnitude in terms of JI comma
>>> interval size as the unison vectors themselves, but since
>>> the angle is very small, this *difference* vector will be
>>> very short (i.e., low complexity). a comma of a given JI
>>> interval size will lead to much higher error if tempered out
>>> over much fewer consonant rungs (i.e., if it's very short)
>>> than if it's tempered out over more consonant rungs (i.e.,
>>> if it's long). therefore, you may end up with a temperament
>>> with much larger error than you would have expected given
>>> your original pair of unison vectors.
>>
>> Right, the difference vector has to vanish, too. Ok. What
>> I don't get is, for a given temperament, can I change the
>> straightness by changing the unison vector representation?
>> If so, this means that badness is not fixed for a given
>> temperament...
>>
>>> that's not true. since both the defining unison vectors *and*
>>> the straightness change, the badness can (and will) remain
>>> constant.
>>
>> Then how can it [the temperament] become "terrible"?
>>
>>> if you change to a "straighter" pair of unison vectors,
>
> Wait a minute -- straightness goes up or down with the angle
> between the vectors? I thought up.
yep -- until you reach 90 degrees (or whatever the equivalent is in
the metric you're using).
>>> one or both of them will have to be a lot shorter, thus less
>>> distribution of error and a worse temperament.
>
> I don't follow the 'shorter' bit. The only thing I thought
> straightness did was make the difference vector more complex.
yes, or if you're decreasing from >90 degrees to 90 degrees, it's the
sum vector that gets more complex.
>> shortening the unison vectors makes the temperament worse, but
>> in a given temperament, this would be counteracted by an
>> increase in straighness, which makes the temperament better.
>
> You lost me.
well, there are probably too many counterfactuals here. why don't we
start again with any examples from the archives which came up in
connection with straightness. your choice.
>> overall, the measure must remain fixed for a given temperament,
>> otherwise it's meaningless.
>
> If by "the measure", you mean badness, I agree.
good!
Message: 6097 - Contents - Hide Contents
Date: Sat, 18 Jan 2003 01:19:14
Subject: Re: heuristic and straightness
From: Carl Lumma
>> > was trying to point out that badness here has failed
>> to reflect your opinion of the temperament.
>
> how so?
You said the temperament got worse, but the badness
remained constant. -C.
Message: 6098 - Contents - Hide Contents
Date: Sat, 18 Jan 2003 01:53:18
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>> I was trying to point out that badness here has failed
>>> to reflect your opinion of the temperament.
>>
>> how so?
>
> You said the temperament got worse,
shortening the unison vectors makes the temperament worse, but in a
given temperament, this would be counteracted by an increase in
straighness, which makes the temperament better. overall, the measure
must remain fixed for a given temperament, otherwise it's meaningless.
Message: 6099 - Contents - Hide Contents
Date: Sat, 18 Jan 2003 05:43:22
Subject: Re: heuristic and straightness
From: Carl Lumma
>> >n most cases the *difference* between the unison vectors
>> will be of similar or greater magnitude in terms of JI comma
>> interval size as the unison vectors themselves, but since
>> the angle is very small, this *difference* vector will be
>> very short (i.e., low complexity). a comma of a given JI
>> interval size will lead to much higher error if tempered out
>> over much fewer consonant rungs (i.e., if it's very short)
>> than if it's tempered out over more consonant rungs (i.e.,
>> if it's long). therefore, you may end up with a temperament
>> with much larger error than you would have expected given
>> your original pair of unison vectors.
>
>Right, the difference vector has to vanish, too. Ok. What
>I don't get is, for a given temperament, can I change the
>straightness by changing the unison vector representation?
>If so, this means that badness is not fixed for a given
>temperament...
>
>> that's not true. since both the defining unison vectors *and*
>> the straightness change, the badness can (and will) remain
>> constant.
>
>Then how can it [the temperament] become "terrible"?
>
>> if you change to a "straighter" pair of unison vectors,
Wait a minute -- straightness goes up or down with the angle
between the vectors? I thought up.
>> one or both of them will have to be a lot shorter, thus less
>> distribution of error and a worse temperament.
I don't follow the 'shorter' bit. The only thing I thought
straightness did was make the difference vector more complex.
>shortening the unison vectors makes the temperament worse, but
>in a given temperament, this would be counteracted by an
>increase in straighness, which makes the temperament better.
You lost me.
>overall, the measure must remain fixed for a given temperament,
>otherwise it's meaningless.
If by "the measure", you mean badness, I agree.
-Carl
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